Mario owns three identical-looking coins. one coin shows heads with probability 1/4, another
shows heads with probability 1/2, and the last shows heads with probability 3/4.
(a) mario randomly picks a coin and flips it. he then picks one of the other two coins and flips it.
let x1 and x2 be the events of the 1st and 2nd flips showing heads, respectively. are x1 and
x2 independent? prove your answer.
(b) mario randomly picks a single coin and flips it twice. let y1 and y2 be the events of the 1st and
2nd flips showing heads, respectively. are y1 and y2 independent? prove your answer.
(c) mario arranges his three coins in a row. he flips the coin on the left, which shows heads. he
then flips the coin in the middle, which shows heads. finally, he flips the coin on the right.
what is the probability that it also shows heads?